6.2 Wiener filtering

In the absence of non-Gaussian sources it is possible to simplify the Maximum Entropy method using
the quadratic approximation to the entropy (Hobson et al.[38]). This has the same affect as chosing a
Gaussian prior so that Equation 16 becomes

where is the covariance matrix of the image vector given by

The solution to the analysis with this Bayesian prior is called Wiener filtering (see, for example, [12]
and [92]) and has been applied to many data sets in the past when non-Gaussianity could be ignored. The
data can be written as

where the convolution of the image vector is with , the beam response of the instrument and
frequency dependance of . is the noise vector. In this case, the best reconstructed image vector,
, is given by

where the Wiener filter, , is given by

and is the noise covariance matrix given by . It is important to note that not only the CMB
signal but all the foregrounds are implicitly assumed to be non-Gaussian in this method ([38][45]
and [37]).